How do you use the Product Rule to find the derivative of #y = x(x^2  2x + 1)^4#?
The product rule allows you to differentiate functions that can be written as the product of two other functions
by using this formula
This means that its derivative can be found by using the product rule
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To find the derivative of ( y = x(x^2  2x + 1)^4 ) using the Product Rule, follow these steps:

Identify the functions ( f(x) ) and ( g(x) ) in the form ( f(x) \cdot g(x) ). ( f(x) = x ) and ( g(x) = (x^2  2x + 1)^4 ).

Apply the Product Rule formula: ( (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x) ).

Find the derivatives of ( f(x) ) and ( g(x) ): ( f'(x) = 1 ) and ( g'(x) = 4(x^2  2x + 1)^3 \cdot (2x  2) ).

Substitute these derivatives into the Product Rule formula: ( y' = 1 \cdot (x^2  2x + 1)^4 + x \cdot 4(x^2  2x + 1)^3 \cdot (2x  2) ).

Simplify the expression: ( y' = (x^2  2x + 1)^4 + 4x(x^2  2x + 1)^3 \cdot (2x  2) ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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